Proceedings of the Twenty-Sixth International Conference on
Automated Planning and Scheduling (ICAPS 2016)
Robust Partial Order Schedules for
RCPSP/max with Durational Uncertainty
Na Fu, Pradeep Varakantham, Hoong Chuin Lau
School of Information Systems, Singapore Management University
80 Stamford Road, Singapore 178902
{nafu, pradeepv, hclau}@smu.edu.sg
to (Fu, Lau, and Varakantham 2015; Bidot et al. 2009;
Lombardi and Milano 2009; Beck and Wilson 2007; Herroelen and Leus 2005). In this paper, we are concerned with
proactive scheduling. Instead of a baseline schedule, we are
interested in a Partial Order Schedule (POS) which is a set of
partially ordered activities such that any embedded temporal
feasible solution is also guaranteed resource feasible (Policella et al. 2009). Within a POS, each activity retains a set
of feasible starting times which provides temporal flexibility
against uncertainty. A simulation based approach to evaluate
the expected makespan of POSs was considered in (Bonfietti, Lombardi, and Milano 2014).
The concrete problem addressed in this paper is Resource
Constrained Project Scheduling Problem with minimum and
maximum time lags (abbrev. RCPSP/max). Extended from
RCPSP, the introduction of temporal separation constraints,
particularly maximum time lags between activities offers a
wide range of modelling capabilities like activity deadlines,
setup times, etc. But it also exemplifies the problem setting
at a much higher level of complexity where the feasibility
problem is already NP-complete (Bartusch, Mohring, and
Radermacher 1988).
Akin to a few existing proactive approaches for considering uncertainty in JSP (Beck and Wilson 2007) and
RCPSP/max (Varakantham, Fu, and Lau 2016; Fu et al.
2012), we consider a risk management objective. Specifically, given a risk parameter α (0 ≤ α ≤ 1) which can be
prescribed by the planner, we are interested in computing
the robust POS which minimizes the α-quantile makespan
distribution. We refer to the least α-quantile also as the αrobust makespan.
In (Fu et al. 2012), heuristic techniques based on local search were provided for generating robust POS for
RCPSP/max under durational uncertainty. However, the limitation of that work is that the derived POS may not meet
maximum temporal constraints for certain uncertainty realisations during execution. This work is motivated by effectively managing risk on temporal constraint violation and
failure to meet robust makespan during POS construction.
Instead of a local search algorithm, we design a Mixed
Integer Linear Programming (MILP) model for POS construction and develop a Benders decomposition algorithm
and heuristic approximation for efficient robust makspan
calculation. In (Varakantham, Fu, and Lau 2016), a start
Abstract
In this work, we consider RCPSP/max with durational uncertainty. We focus on computing robust Partial Order Schedules (or, in short POS) which can be executed with risk
controlled feasibility and optimality, i.e., there is stochastic posteriori quality guarantee that the derived POS can be
executed with all constraints honored and completion before robust makespan. To address this problem, we propose
BACCHUS: a solution method on Benders Accelerated Cut
Creation for Handling Uncertainty in Scheduling. In our proposed approach, we first give an MILP formulation for the
deterministic RCPSP/max and partition the model into POS
generation process and start time schedule determination.
Then we develop Benders algorithm and propose cut generation scheme designed for effective convergence to optimality for RCPSP/max. To account for durational uncertainty,
we extend the deterministic model by additional consideration of duration scenarios. In the extended MILP, the risks
of constraint violation and failure to meet robust makespan
are counted during POS exploration. We then approximate
the uncertainty problem with computing a risk value related
percentile of activity durations from the uncertainty distributions. Finally, we apply Pareto cut generation scheme and
propose heuristics for infeasibility cuts to accelerate the algorithm process. Experimental results demonstrate that BACCHUS efficiently and effectively generates robust solutions
for scheduling under uncertainty.
Introduction
Most research on project scheduling focus on a perfectly
pre-defined scheduling environment. However, durational
uncertainty in real world projects may always happen and
make deadline-driven project management a challenging
and difficult task. Thus, effectively handling durational uncertainty in project scheduling is of realistic and practical
importance.
Broadly, one may classify the decision-making approaches for tackling uncertainty in scheduling into two categories: Proactive scheduling computes an apriori buffered
schedule or policy before uncertainty occur. Reactive
scheduling provides online decisions on starting next activity when uncertainty occurs. For a survey of existing
works on project scheduling uncertainty, one may refer
c 2016, Association for the Advancement of Artificial
Copyright Intelligence (www.aaai.org). All rights reserved.
124
time schedule with robust makespan was explored through a
proactive sampling based technique. In that work, the probability of failure is counted when resource capacity constraints are violated. The characteristic feature of our proposed algorithm is the effort to effectively account for risk
controlled feasibility and solution quality in terms of robust
makespan. Thus, we manage the risks when temporal constraint is violated, and/or the actual execution fails to meet
the robust makespan.
More specifically, we propose an approach, Benders
Accelerated Cut Creation for Handling Uncertainty (BACCHUS). BACCHUS consists of five components:
• An MILP formulation for the deterministic RCPSP/max
and partition of POS generation process from scheduling
decision making.
• An effective cut generation mechanism explored for resolving constraints conflicts used in Benders decomposition algorithm.
• An extension of the exact MILP model for solving the
deterministic problem with sampling approximation for
accommodating additional durational uncertainty.
• A scalable solution to robust optimization for solving
RCPSP/max with durational uncertainty.
• Cut generation enhancements during iteration process for
accelerating Benders algorithm.
min vy
y
s.t. P r((vy (p̃) ≥ vy ) ∨ (s > T)) ≤ α
Table 1: BACCHUS: Optimization Model
of activity ai . The goal of the deterministic RCPSP/max
is to determine a feasible schedule, such that the project
makespan, which is defined as the start time of the final
dummy activity aN +1 , is minimized.
RCPSP/max with Durational Uncertainty
In this work, we consider project scheduling under uncertainty, where the stochastic characteristics apply to the activity durations. In the deterministic setting, makespan can
be used to evaluate the performance of a project. However, when uncertainty is involved, the makespan itself becomes a random variable. Similar to existing work (Fu et al.
2012), we employ the metric of α-robust makespan as the
project objective. As indicated earlier, α-robust makespan
for a scheduling project can be defined as the minimum αquantile value over all possible makespan distributions of
POSs.
Formally, given a value of risk α (0 ≤ α ≤ 1), our goal
is to find a POS y with the makespan achieving the value of
α-robust makespan. Let vy represent the α-quantile of the
makespan distribution represented by POS y, i.e.,
The Deterministic RCPSP/max
The RCPSP/max consists of N activities {a1 , · · · , aN } and
K types of renewable resources limited by capacity Ck ,
where k = 1, · · · , K. Each activity ai requires rik units of
resources of type k to be executed for a duration of pi time
units without preemption.
Generalized temporal constraints for RCPSP/max can
specify a minimal or maximal time lag between any pair of
min
specifies that activity j
activities. A minimal time lag Ti,j
can only be started (or finished) when activity i has already
min
.A
started (or finished) for a certain time period of Ti,j
max
specifies that activity j should be
maximal time lag Ti,j
max
started (or finished) at the latest a certain number of Ti,j
time units beyond the start (or finish) of activity i. Thus,
there exist four types of generalized temporal constraints:
start-start, start-finish, finish-start and finish-finish. In the
deterministic setting, the different types of constraints can
be represented in standardized start-start form by using the
transformation rules (Bartusch, Mohring, and Radermacher
1988).
The two types of constraints involved in the RCPSP/max
can be summarized as follows:
• Generalized Temporal Constraints (s ≤ T):
P r((vy (p̃) ≥ vy ) ∨ (s > T)) ≤ α
where vy (p̃) is a random variable that denotes the makespan
distribution represented by POS y and p̃ denotes the uncertain activity durations. Therefore, the α-robust makespan v ∗ ,
which is also the least value of vy over all possible makespan
distributions, can be computed by solving the robust optimization problem in Table 1.
Solving the Deterministic RCPSP/max
Inspired by the work in (Artigues, Michelon, and Reusser
2003) on a flow-based continuous time formulation for
RCPSP, we give an MILP formulation for RCPSP/max for
POS construction and schedule determination. This model
would be extended to handle durational uncertainty in the
stochastic scheduling problem presented in the following
section.
The Model
We define the following decision variables:
• xkij : resource flow variables representing the number of
resource units of type k transferred directly from activity
ai to activity aj .
min
max
≤ sj − si ≤ Ti,j
, ∀i, j.
Ti,j
• Resource Capacity Constraints:
rik ≤ Ck , ∀t, k.
• yij : sequencing variables used for POS construction,
yij = 1 if and only if activity ai precedes activity aj .
{i|si ≤t≤si +pi }
A schedule S = (s1 , · · · , sN ) is an assignment of start
times to all activities, where si represents the start time
• si : scheduling variables for determining the start time of
activity ai .
125
v = min sn+1
M aster − M ILP (ρ){
s.t. sj − si ≥ Tijmin ∀(i, j) ∈ T min
sj − si ≤ Tijmax , ∀(i, j) ∈ T max
(1)
(2)
sj ≥ si + p0i − M (1 − yij ) ∀i, j
si ≥ 0 ∀i
(3)
(4)
xkij ≤ min{rik , rjk }yij ∀i = 0, j = n + 1, k
(5)
xk0j
≤ rjk ∀j, k
xkin+1 ≤ rik ∀i, k
k
k
xij =
xji = rik ∀i = 0, n + 1, k
j
j
min z
s.t. Constraints 5 − 12
z ≥ ατ (y) ∀τ = 1, ...ρ
βτ (y) ≥ 0 ∀τ = 1, ...ρ
}
(6)
(7)
Table 3: The Master Problem
(8)
j
xk0j
=
xkjn+1 ≤ Ck ∀k
v(ȳρ ) = min sn+1
(9)
s.t. sj − si ≥ Tijmin ∀(i, j) ∈ T min
j
xkij
≥ 0, ∀i, j, k
yij ∈ {0, 1} ∀i, j
yij + yji ≤ 1 ∀i, j
(13)
(14)
sj − si ≤ Tijmax , ∀(i, j) ∈ T max
(10)
(11)
(12)
sj ≥ si +
p0i
− M (1 −
ρ
ȳij
)
∀i, j
si ≥ 0 ∀i
Table 2: RCPSPMax(p0 )
(15)
(16)
(17)
(18)
Table 4: The Slave Problem
Table 2 presents the MILP formulation. Constraints 1 and
2 express the generalized temporal constraints with minimum and maximum time lags between the starting times of
two activities. Constraint 3 links the starting times of activities ai and aj with sequencing variables yij . In other words,
the constraint constructs POS in terms of yij . It is active
when yij = 1 which enforces the precedence relationship
sj ≥ si + p0i . Otherwise, the constraint is always satisfied if
there is no precedence relationship between ai and aj , i.e.,
yij = 0. In that case, no resource flows are carried from ai to
aj which sets xkij = 0. But if ai precedes aj , the maximum
resource flow sent ai to aj is forced to be min{rik , rjk },
as shown in Constraint 5. The constraint also expresses that
if there is a positive resource flow transfer from ai to aj ,
i.e., xkij > 0, then the precedence relation is enforced with
yij = 1. Constraints 6 and 7 handle boundary conditions.
Constraints 8-10 are flow conservation constraints where
Constraint 8 states that the total flows sent to and from nondummy activity ai equal to its resource requirement of the
corresponding resource type, rik . The total resource of type
k for dispatching into the project network from the starting
dummy node a0 and collected at the sink dummy node an+1
is upper bounded by its capacity, Ck as represented in Constraint 9. Constraints 11 and 12 are constituted for POS construction. Constraint 12 covers the total three relationships
between two activities, either ai precedes aj , or aj precedes
ai , or ai and aj are executed in parallel.
composition algorithm for solving RCPSP/max.
In the MILP formulation of Table 2, there are three types
of decision variables. The flow variable x and starting time
variables s are continuous, but the sequencing variables y
are complicating in taking binary integers. To provide an efficient solution method for solving large scale RCPSP/max,
we partition those decision variables into two sets, (x&y, s).
x&y are involved in the master problem where a resource
feasible precedence decision rule is generated. For a fixed
decision rule, the slave focuses on generating the optimal
schedule by deciding the starting times s.
At each iteration, we solve a relaxed master problem and
generate a resource feasible POS. For a fixed POS y, the
slave problem decides starting time schedule with the goal of
minimizing the project makespan. The Master problem and
the Slave problem at the ρth iteration are given in Table 3
and Table 4, respectively. An each iteration, new cuts (to
be explained in next section) as in Constraint 13 (optimality cuts) and Constraint 14 (global cuts and feasibility cuts)
are added to the master problem and then make it progress
towards an optimal solution.
Cut Generation Scheme
The way of generating effective cuts during iteration process is always the key to the success of a Benders decomposition algorithm. In this work, we developed three types of
cuts for guiding master to generate good candidate POSs for
converging to optimality: Global Cuts, Optimality Cuts and
Feasibility Cuts.
Benders Decomposition Algorithm
Benders Decomposition (Benders 1962) is a solution approach for large scale combinatorial optimization problem,
based on the idea of partition and cut generation. One may
refer to (Li and Womer 2009) and (Costa 2005) for its successful application in solving multi-skilled personnel and
network design problem. In this work, we developed this de-
Global Cuts
We first explore the static global cuts before the main algorithm iterations start. The main feature of our proposed cuts
generation scheme is to explore problem structures and build
constraints to avoid conflicting assignments to the sequenc-
126
ing variables y. The role of solving the slave problem is to
trigger the violated constraints and infer the cuts to the master problem. In the following, we infer cuts from the causes
of infeasibility by temporal analysis and precedence transitivity analysis.
max
(Tijmin λij − Tijmax μij + (p0i − M (1 − ȳij ))γij )
i,j!=i
s.t.
(λij − λji − μij + μji + γij − γji ) ≥ 0 ∀i (19)
j|j=i,i=n+1
Temporal Analysis For activities ai , aj and am , we first
update the existing temporal constraints between activities
by using the following formulas:
(λij − λji − μij + μji + γij − γji ) ≥ −1
(20)
j|j=i,i=n+1
λij , μij , γij ≥ 0, ∀i, j, i = j
(21)
max
max
max
max
• Tijmax ≥ Tim
+Tmj
=⇒ Tijmax = Tim
+Tmj
∀i, j, m
min
min
min
min
+ Tmj
=⇒ Tijmin = Tim
+ Tmj
∀i, j, m
• Tijmin ≤ Tim
Table 5: The Slave Dual
max
min
max
min
• Tijmax ≥ Tmj
−Tmi
=⇒ Tijmax = Tmj
−Tmi
∀i, j, m
min
max
min
max
• Tijmin ≤ Tmj
− Tmi
=⇒ Tijmin = Tmj
− Tmi
∀i, j, m
Precedence Transitivity Analysis
• yim ≥ yij + yjm − 1 ∀i, j, m
The idea is, to reconstruct the temporal time lags between
activities if tighter lags can be inferred. Based on the temporal analysis designed for resolving resource conflicts, we
have the following observations and such constraints must
be satisfied for all feasible solutions.
The set of cuts are used to capture the transitivity of the
ordering relations: if ai precedes aj and aj precedes am ,
then we have ai precedes am .
Optimality Cuts
• Tijmin ≥ p0i =⇒ yij = 1 ∀i, j
For a fixed policy y, if the slave is feasible, the minimization
objective achieved at that iteration would be an upper bound
to the optimization problem. Optimality cuts can then be derived and added to the master in next iteration to improve
the approximation of the master objective from the optimal
solution.
To examine the slave MILP model, we introduce nonnegative dual variables λij , μij and γij for Constraints 15,
16, and 17, respectively. Table 5 shows the dual LP of the
slave problem. Suppose that the primal LP in Table 4 is always feasible for every policy y chosen, then the dual LP
has a bounded feasible region. Let set U contain the extreme
points of the polyhedron D defined by the constraints of Table 5 then we can instantiate Constraint 13 in the master
by updating optimal cuts as provided in the following constraint:
For activities ai and aj , if the minimum time lags Tijmin
is no less than the duration of activity i, ai must be executed
before aj to guarantee the solution feasibility. For this case,
we add the cut yij = 1.
• Tijmax < p0i =⇒ yij = 0 ∀i, j
If the maximum time lag Tijmax is less than the duration of
activity ai , then aj cannot be executed after ai and we set
yij = 0.
• ∃ resource k, rik + rjk > Ck =⇒ yij + yji = 1 ∀i, j, k
If there exists a resource type k, of which the overall consumption of activities ai and aj exceeds the capacity Ck ,
then both activities cannot be executed in parallel in order to respect resource constraints. In such a case, either
ai precedes aj or aj precedes ai . Thus, we add the cut
yij + yji = 1 to remove the parallel case.
•
p0i
+
p0m
>
max
Tin
+
max
Tmj
αρ (y) =
ρ
(Tijmin λρij − Tijmax μρij + (p0i − M (1 − yij ))γij
)
i,j=i
(22)
=⇒ yij + ymn ≤ 1 ∀i, j, m, n
where (λ, μ, γ) ∈ U .
Every time a slave generates an optimal solution, an optimal cut can be constructed in terms of the dual values of the
slave. Such constraints are then added to the master problem
to improve the lower bound of the optimization problem.
Proof. Suppose yij + ymn > 1, i.e., yij = ymn = 1, we then
have sj ≥ si + p0i and sn ≥ sm + p0m . The summation of
these two inequations imply that (sn − si ) + (sj − sm ) ≥
p0i + p0m . Given the definition of temporal constraints, we
max
max
+ Tmj
≥ p0i + p0m . Hence proved.
then have Tin
This type of cut resolves the problem that pairs of activities cannot be sequentially connected simultaneously due to
relationship between activity durations and time lags.
Feasibility Cuts
However, it is possible that during iterating process, the generated POS from a relaxed master cannot produce a feasible
schedule in the slave model. The feasibility cuts need to be
inferred to deal with such cases. The feasibility cuts for iteration ρ have the general form in the following constraint:
max
min
• p0i + p0m > Tin
− Tjm
=⇒ yij + ymn ≤ 1 ∀i, j, m, n
As above.
• Tijmax < p0i + p0m , =⇒ yim + ymj ≤ 1 ∀i, j, m
Proof. Suppose yim + ymj > 1, i.e., yim = ymj = 1, we
then have sm ≥ si + p0i and sj ≥ sm + p0m . The summation
of these two inequations imply that sj −si ≥ p0i +p0m . Given
sj − si ≤ Tijmax , we then obtain Tijmax ≥ p0i + p0m . Hence
proved.
ρ
(1 − yij ) +
i,j|ȳij =1
yij ≥ 1.
(23)
ρ
i,j|ȳij =0
The idea is that, once infeasibility occurs at POS ȳ ρ , the
total number of binary variables flipping their value with respect to ȳ ρ either from 1 to 0 or from 0 to 1 is at least 1.
127
Let U B and LB represent the upper bound and the lower
bound of the optimal solution, respectively. Algorithm 1 provides pseudocode of the extended benders decomposition algorithm for RCPSP/max.
min RM
s.t. Constraints 5 − 12
sqj ≥ sqi + pqi − M (1 − yij ) ∀i, j, q
sqj
sqi
sqj
Algorithm 1: Extended Benders Decomposition Algorithm (RCPSP/max Instance)
1: LB ← 0, U B ← ∞, ρ ← 0
2: Temporal analysis to obtain global cuts in
Constraint 14;
3: terminate ← f alse
4: while terminate = f alse do
5:
Solve master problem and obtain ȳρ ;
6:
if z(ρ)= UB then
7:
terminate ← true;
8:
else
9:
LB ← z(ρ);
10:
Update constraints in the Slave problem;
11:
Solve the slave problem;
12:
if The slave problem is feasible then
13:
Update UB;
14:
Generate optimality cuts in Constraint 13;
15:
else
16:
Generate feasibility cuts in Constraint 14;
17:
end if
18:
Add cuts to master problem;
19:
end if
20:
ρ←ρ+1
21: end while
−
sqi
≥
Tijmin
∀(i, j) ∈ T
min
,q
≥ 0 ∀i, q
− sqi ≤ Tijmax + z q M ∀i, j, q
q
z ≤ αQ ∀q
(24)
(25)
(26)
(27)
(28)
q
M z q > sqn+1 − RM ∀q
(29)
Table 6: RCPSPMaxUnc ({pq }q=1,...Q )
the robust makespan RM .
Though a very useful model for representing uncertainty
in durations, as the problem scale increases, the dependency on number of samples employed gets the scalability issue of the multi-sample model worsened and it can
hardly find solutions with only 20 samples adopted for
J20 and J30 instances within a 10 minute time limit. To
overcome the scalability problem, we adopt the idea of
percentile sample approximation as in (Varakantham, Fu,
and Lau 2016) and summarize the scenario sample set by
using one (1 − α)-percentile duration sample. The duration of activity ai in the percentile sample can be presented as p̂i = P ERCEN T ILE1−α (pi ), where pi =
q
{p1i , · · · , pqi , · · · pQ
i } and pi is the duration of activity ai in
sample q. The idea is to entail that at least in (1 − α) · Q
number of samples, duration of activity ai is lower than p̂i .
Therefore, the stochastic model in Table 6 can be summarized in RCPSPMaxUnc (p̂) with the implementation to be
shown in next section.
Solving RCPSP/max with Durational
Uncertainty
Let sqi represent the scheduling variable for determining the
start time of activity ai on sample q, where q = 1, ...Q and
Q is the total number of samples. The indicator variable zq
equal to 1 if and only if the constructed POS cannot be executed feasibly or efficiently on sample q. As POS is generated from the master problem where resource feasibility
were accommodated during construction, the temporal aspect is the only concern for feasibility. With the objective of
robust optimization problem in Table 1 being generating a
POS with the best robust makespan value, in the stochastic
model of Table 6, we consider both feasibility and efficiency
properties during POS exploration.
Specifically, Constraints 27-29 control the potential probability of failure during the POS construction. As the introduction of resource flow variables would already guarantee resource feasibility during schedule generation, zq = 0
enforces the temporal constraints of maximum time lags,
thus, it guarantees the schedule feasibility, as shown in Constraint 27. Let α represent the risk threshold that can be
prescribed by the planner. Constraint 28 measures solution
quality in terms of successful execution on Q different scenarios of realized uncertainty, and Constraint 29 considers
the makespan for performance evaluation. zq = 1 if the realized makespan of sample q termed as sqn+1 reaches beyond
Experimental Results
In this section, we conducted computational experiments
to test the performance of BACCHUS and compare with
the existing best known approaches SORU-H (Varakantham,
Fu, and Lau 2016) and FPVL (Fu et al. 2012) for generating
α-robust makepan for RCPSP/max under durational uncertainty.
The problem instances we run BACCHUS on are
extended from benchmark sets J10 J20 and J30 for
RCPSP/max from PSPlib (Kolisch, Schwindt, and Sprecher
1998). Each data set contains 270 instances and each instance has 5 types of resources available. The number of
activities for instances in J10, J20 and J30 are 10, 20 and
30, respectively. We assume the processing time pi of activity ai is normally distributed with mean value corresponding
to the deterministic duration p0i given by benchmarks and
standard deviation denoted as σ. The duration scenarios are
generated by sampling from normal distributions with three
different duration variabilities σ = {0.1, 0.5, 1}. We implemented BACCHUS in Java and Cplex studio on a Core
i7-4790 CPU 3.60GHz processor with 32.0 GB RAM and
128
slave problem at current iteration. The objective maximizes
strength of the cut for relative interior y 0 of the feasible
space defined in master constraints, and constraints of the
model specify the feasible space of the slave problem. We
implement Pareto optimal cut generation scheme on large
scale instances of J20 and J30 and observe an average reduction of 18.6% and 23.9% of running time required to solve
to optimality, respectively.
When the slave problem becomes infeasible during iterating process, an alternative way is to change sequencing decisions on a (a > 1) pairs of activities simultaneously with
respect to the current POS ȳ generated, rather than only one
pair of activities, as of the RHS parameter of Equation 23.
In other words, the feasibility cut can be improved as
Robust Makespan VS Risk
Robust Makespan
52
50
48
46
44
42
40
0.05
0.1
0.2
Risk
Standard Deviation = 0.1
Standard Deviation = 0.5
Standard Deviation = 1
Figure 1: Robust Makespan VS Risk α and Standard Deviation σ
max
ρ
(1 − yij ) +
i,j|ȳij =1
yij ≥ a.
(30)
ρ
i,j|ȳij =0
The motivation for varying step size a is that, for large scale
problem instances, restricting only one sequencing decision
to be revised at each iteration may not be efficient enough to
restore feasibility. We vary the value of a from 1 to 5, and
observe that the number of infeasible iterations for almost all
instances get reduced to some extent. But there is no strict
monotonicity between the running time and values of a. To
show this trend, we randomly pick one instance from each
data set and report the running time in logarithm value with
different a values in Figure 2. We observe that, as the step
size slightly increases from 1, the running times required for
all instances decreases obviously. But as a increases, there
seems exist a threshold where step size taking higher values
would bring negative effect on algorithm efficiency. Unfortunately, we are not able to find a universal optimal value
of a for best algorithm performance, because it is highly instance dependent. Note that in Figure 2, the running time
was solved optimality for all instances, except for J10 instance at a = 5, where optimality is comprised and higher
robust makespan is returned. One direct insight from our observation is that, a large step size a may be over-corrected
for restoring feasibility during benders iterations, especially
for small scale problems. But it would still be promising
to design a good step size value for improving algorithm
efficiency for specific problem, especially when the size is
large.
Now we compare BACCHUS with the existing approaches SORU-H (Varakantham, Fu, and Lau 2016) and
FPVL (Fu et al. 2012) to solving RCPSP/max with durational uncertainty. All three works aim for exploring αrobust makepan. Thus, a straightforward metric to perform
a computational comparison is on the value of α-robust
makespan. Table 8 summaries the average values of α-robust
makespan obtained from BACCHUS, FPVL, SORU-H over
270 instances of all three benchmark sets for α = 0.1 and
σ = 0.5. We first compare values of robust makespan with
FPVL as both solutions returned in BACCHUS and FPVL
are POSs. The experimental results show that BACCHUS
outperforms FLPV in generating more robust POSs.
Instead of scheduling policy exploration, SORU-H focuses on generating the start time schedule. Though BACCHUS has the same robust makespan objective, the ways of
0
(Tijmin λij − Tijmax μij + (p0i − M (1 − ȳij
))γij )
i,j!=i
s.t.
Constraints 19 − 21
min
v(ȳ) =
(Tij λij − Tijmax μij + (p0i − M (1 − ȳij ))γij )
i,j!=i
Table 7: Pareto optimal Cuts Generation
average the results over 10 runs for each problem instance.
We first run experiments on three data sets across increasing levels of risk ε = {0.05, 0.1, 0.2} and varying standard
deviations. For open problem instances that cannot be solved
to optimality, we report the best results obtained within a 10minute time limit. Figure 1 summarizes the average values
of robust makespans over 270 instances in J10 with varying α and ε. We observe that, the value of robust makespan
increases as the level of risk decreases. In other words, the
lower risk value the decision maker is willing to take, the
higher robust makespan for the POS generated. The lower
degree of durational perturbation, the better value of robust
makespan obtained.
Then, we explore heuristic enhancements for generating effective cuts and improving the algorithm efficiency.
During the procedure of Benders iterations, it may happen that the slave solution is not unique, especially when
the slave optimization suffers from degeneracy. To reduce
the number of iterations towards convergence during Benders processing, the impact of stronger cuts were examined
in (T. L. Magnanti 1981). In our experiments, we adopt
the same idea and implement Pareto optimal cut generation for exploring stronger cuts. At iteration ρ, a cut generated from extreme point (λρ1 , μρ1 , γ ρ1 ) is said to dominate a cut generated from extreme point (λρ2 , μρ2 , γ ρ2 ),
if αρ (ȳ, λρ1 , μρ1 , γ ρ1 ) ≥ αρ (ȳ, λρ2 , μρ2 , γ ρ2 ) defined in
Equation 22 with strict inequality at least one ȳ. A cut
dominated by no other cuts is said to be Pareto optimal. A
Pareto optimal cut can be generated by solving the model
in Table 7, where v(ȳ) is the optimal solution value of the
129
ter robust makepan. This is also the first work that potential
risk on temporal constraint violations are proactively managed in POS construction to hedge against uncertainty.
13
Running Time (ln)
12
11
10
Acknowledgements
J10
This research is supported by Singapore National Research
Foundation under its International Research Centre @ Singapore Funding Initiative and administered by the IDM Programme Office, Media Development Authority (MDA).
J20
9
J30
8
7
1
2
3
4
References
5
Step Size a for Infeasibility Cuts in Constraint 30
Artigues, C.; Michelon, P.; and Reusser, S. 2003. Insertion techniques for static and dynamic resource constrained project scheduling. European Journal of Operational Research 149:249–267.
Bartusch, M.; Mohring, R. H.; and Radermacher, F. J. 1988.
Scheduling project networks with resource constraints and time
windows. Annals of Operations Research 16(1-4):201–240.
Beck, J. C., and Wilson, N. 2007. Proactive algorithms for job
shop scheduling with probabilistic durations. Journal of Artificial
Intelligence Research 28(1):183–232.
Benders, J. 1962. Partitioning procedures for solving mixedvariables programming problems. Numerische Mathematik 4:238–
252.
Bidot, J.; Vidal, T.; Laborie, P.; and Beck, J. C. 2009. A theoretic and practical framework for scheduling in a stochastic environment. Journal of Scheduling 12:315–344.
Bonfietti, A.; Lombardi, M.; and Milano, M. 2014. Disregarding
duration uncertainty in partial order schedules? yes, we can! In
Integration of AI and OR Techniques in Constraint Programming 11th International Conference, CPAIOR, 210–225.
Costa, A. M. 2005. A survey on benders decomposition applied to fixed-charge network design problems. Comput. Oper. Res.
32(6):1429–1450.
Fu, N.; Lau, H. C.; Varakantham, P.; and Xiao, F. 2012. Robust
local search for solving rcpsp/max with durational uncertainty. J.
Artif. Intell. Res. (JAIR) 43:43–86.
Fu, N.; Lau, H. C.; and Varakantham, P. 2015. Robust execution strategies for project scheduling with unreliable resources and
stochastic durations. J. Scheduling 18(6):607–622.
Herroelen, W., and Leus, R. 2005. Project scheduling under uncertainty: Survey and research potentials. In European Journal of
Operational Research, volume 165(2), 289–306.
Kolisch, R.; Schwindt, C.; and Sprecher, A. 1998. Benchmark Instances for Project Scheduling Problems. Kluwer Academic Publishers, Boston. 197–212.
Li, H., and Womer, K. 2009. Scheduling projects with multi-skilled
personnel by a hybrid milp/cp benders decomposition algorithm. J.
Scheduling 12(3):281–298.
Lombardi, M., and Milano, M. 2009. A precedence constraint posting approach for the rcpsp with time lags and variable durations. In
Proceedings of the 15th international conference on Principles and
practice of constraint programming, CP’09, 569–583.
Policella, N.; Cesta, A.; Oddi, A.; and Smith, S. 2009. Solve-androbustify. Journal of Scheduling 12:299–314.
T. L. Magnanti, R. T. W. 1981. Accelerating benders decomposition: Algorithmic enhancement and model selection criteria. Operations Research 29(3):464–484.
Varakantham, P.; Fu, N.; and Lau, H. C. 2016. A proactive sampling approach to project scheduling under uncertainty. In AAAI.
Figure 2: Effect of Step Size a for Infeasibility Cuts on Algorithm Efficiency
J10
J20
J30
BACCHUS
47.26
82.34
107.27
FLPV
51.3
84.6
113.3
SORU-H
48.43
79.8
103.2
Table 8: Comparison with FLPV and SORU-H on α-robust
makespan
tackling uncertainty in BACCHUS and SORU-H are totally
different. In SORU-H, temporal constraints are always honored and resource violation is the only concern for schedule infeasibility. BACCHUS solves instead hard resource
constraints and soft temporal constraints. Given the difference of solution results and approaches between SORU-H
and BACCHUS, robust makespan may not be a proper index for making comparison of solution performances. A
promising direction of our future work is to implement the
resulted solutions from BACCHUS and SORU-H in real
world scheduling environment and evaluate the performance
by examining the real probability of failure under different uncertainty scenarios. Thus, though the values of robust
makespan for BACCHUS is slightly higher than SORU-H
for J20 and J30, we still believe that BACCHUS can provide
an alternative solution in the form of more flexible POSs
for decision makers to efficiently tacking project scheduling
problem under uncertainty.
Conclusion
In this work, we proposed BACCHUS: a solution method
for handling uncertainty in scheduling, consisting of four
phases: 1) modeling RCPSP/max in the form of MILP
and partition into the POS determination and the start time
scheduling process; 2) developing Benders decomposition
algorithm and proposing cut generation scheme by exploring the problem structure of RCPSP/max; 3) computing for
robust POSs that can be executed within risk controlled performance guarantee of feasibility and optimality; 4) accelerating Benders iterations from heuristic based techniques.
Experimental results demonstrate BACCHUS outperforms
the best approach on robust POS generation by deriving bet-
130